A generalization of Drewnowski s result on the Cantor-Bernstein type theorem for a class of nonseparable Banach spaces

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1 A generalization of Drewnowski s result on the Cantor-Bernstein type theorem for a class of nonseparable Banach spaces Marcos J. González Marek Wójtowicz Abstract Let X a denote the order continuous part of a Banach lattice X, and let Γ be an uncountable set. We extend Drewnowski s theorem on the comparison of linear dimensions between Banach spaces having uncountable symmetric bases to a class of discrete Banach lattices, the so-called D-spaces. We show that if X and Y are two D-spaces and there are continuous linear injections (not necessarily embeddings) from X into Y and vice versa, then X and Y are order-topologically isomorphic. In the proof we apply a theorem on the extension of an order isomorphism from X a onto Y a to an order isomorphism from X onto Y, the classical Drewnowski s theorem, and a supplement of Troyanski s theorem on embeddings of l 1(Γ) spaces into a Banach space with an uncountable symmetric basis. Our result applies to the class of Orlicz spaces l ϕ(γ), where ϕ is an Orlicz function. Keywords: Linear dimension, Discrete Banach lattice, Levi norm, Uncountable symmetric basis, Orlicz space MSC: 46B26, 46B42, 46B45, 47B37. 1 Introduction We use standard notations and for notions undefined below we refer the reader to the next section and to the monographs [4, 19, 23]. In what follows, Γ and Θ denote fixed uncountable sets, all Banach spaces are of infinite dimension, and all operators are linear and continuous. For X a Banach lattice, the symbol X a stands for the order continuous part of X. Following Banach [5, Chap. XII, p. 193], for two Banach spaces X and Y, we write dim l (X) dim l (Y ) if X embeds isomorphically into Y. If both dim l (X) dim l (Y ) and dim l (Y ) dim l (X), we write dim l (X) = dim l (Y ) and say that X and Y have the same linear dimension. 1

2 In this paper, we consider theorems of the following form, which we call Cantor-Bernstein type theorems: Let B be a fixed class of Banach spaces, and let X, Y B be such that there are continuous linear injections from X into Y and from Y into X (in particular, this is so if dim l (X) = dim l (Y )). Then X and Y are isomorphic. In sections 3, 4, and 5 we extend remarkable results obtained in 1987 by L. Drewnowski (Propositions 1.2 and 1.4 below) concerning the comparison of linear dimensions between nonseparable Banach spaces having uncountable symmetric bases. Notice that, if X is such a real space, it can be considered as a discrete Banach lattice with X = X a (see [16, p. 2]). In our main result, stated in Theorem 4.2, we show that the series of the equivalences in the Drewnowski s theorems remain true also in a large class of discrete Banach lattices X with X X a. In 1932, Banach and Mazur [6] proved that dim l (C[0, 1]) = dim l (C[0, 1] l 1 ), yet the spaces C[0, 1] and C[0, 1] l 1 are not isomorphic because dim l (C[0, 1] ) dim l ((C[0, 1] l 1 ) ). It is interesting to note that Banach obtained the following positive result [5, Lemme and Théorème 4 6 on pp ]: Proposition 1.1. If p, q (1, ), then dim l (L p [0, 1]) = dim l (L q [0, 1]) if and only if p = q. (Cf. [18]; see also [14] for details of the many contributors to this achievement and an account of its proof.) In 1987, Drewnowski [8, Theorem] extended the above Banach s result to a class of nonseparable Banach spaces as follows: Proposition 1.2. Let X and Y be nonseparable Banach spaces having uncountable symmetric bases (x γ ) γ Γ and (y t ) t Θ, respectively. Then the following conditions are equivalent: (i) X and Y have the same linear dimension. (ii) X and Y are isomorphic. (iii) There is a bijection f : Γ Θ and an isomorphism T : X Y such that T (x γ ) = y f(γ), for every γ Γ. The proof uses the result below, due to Troyanski [20, Corollary 2], which, roughly speaking, asserts that l 1 -bases have maximal linear dimension in the family of uncountable symmetric bases (recall that a symmetric basis (x γ ) γ Γ for some Banach space is said to be an l 1 -basis, whenever the basis (x γ ) γ Γ is equivalent to the unit vector basis (e γ ) γ Γ of l 1 (Γ)). Proposition 1.3. Let X be a (nonseparable) Banach space with an uncountable symmetric basis (x t ) t Θ. If X contains an isomorphic copy of l 1 (Γ), then (x t ) is an l 1 -basis. 2

3 (Note that a similar result [20, Corollary 1] holds for c 0 (Γ) instead of l 1 (Γ). See also [9, 10] and the monograph [11] for other results in this direction.) In the same paper [8], Drewnowski observes that in some cases the assumptions in Proposition 1.2 can be weakened: he indicates how to adapt its proof to obtain the following result [8, Corollary on p. 161], which we extend in Theorem 4.2 to a larger class of Banach spaces: Proposition 1.4. Let X and Y be nonseparable Banach spaces having uncountable symmetric bases (x γ ) γ Γ and (y t ) t Θ, respectively. Assume further that both the bases are non-l 1 bases. Then conditions (i) (iii) in Proposition 1.2 are equivalent to the condition (iv) There are continuous linear injections X Y and Y X. This result readily applies to the class of Orlicz spaces l ϕ (Γ), where ϕ is an Orlicz function fulfilling the 2 -condition at zero. Then the space l ϕ (Γ) = (l ϕ (Γ)) a has an uncountable symmetric basis, and the basis is an l 1 -basis if and only if ϕ is equivalent at zero to the identity. From our Theorem 4.2 it follows that the equivalence (i) (ii) in Proposition 1.2 holds true in the whole class of (real) spaces of the form l ϕ (Γ), where card(γ) 2 ℵ0 and the Orlicz function ϕ is arbitrary (see Section 5). Here one should note that there are separable Orlicz sequence spaces with uncountably non-equivalent symmetric bases [15, Theorem 4.b.9]. This rules out the equivalence of conditions (ii) and (iii) in Proposition 1.2 for the separable case. On the other hand, as far as we know, it is an open question of whether or not conditions (i) and (ii) in Proposition 1.2 are equivalent for sequence spaces with a symmetric basis. The paper is organized as follows. In the next section, we fix notation and present the basic facts needed for the proof of our main results. In Section 3, we address the critical l 1 -case : (*) in Theorem 3.1 we supplement Troyanski s Proposition 1.3, changing the role of the spaces involved and considering continuous injections instead of isomorphic embeddings, and (**) in Theorem 3.3 we show that, for every pair G, D of infinite subsets of the interval [0, 1], there exist a continuous injection from l 1 (D) into l 1 (G). These results suggest it makes sense to restrict our studies to (nonseparable) discrete Banach lattices with a set of atoms of the cardinality at least continuum. In Section 4, we introduce the notion of a D-space and prove our main theorem for such spaces. In Section 5 we illustrate how our results work for a class of big discrete Orlicz spaces. 2 Preliminaries Throughout this section, X denotes either an Archimedean vector lattice or a Banach space; in the latter case denotes a norm on X, and if, additionally, X is a Banach lattice, the norm is a lattice norm (i.e., the condition x y implies that x y ; here the symbol x denotes the modulus of x X). 3

4 Let X be a Banach space, and let J be an infinite set. A family (x j ) j J of elements in X is said to be an unconditional basis of X if, for every x X there is a unique family of scalars (a j ) j J such that the series x = a j x j ( ) j J converges unconditionally in X, i.e., for every ε > 0 there is a finite subset F of J such that x j F a jx j ε for every finite subset F of J such that F F. In particular, for every x X, at most countably many coordinates a j are nonzero. Then the support of x X is defined as s(x) := {j J : a j 0}, and s(x) is a countable subset of J. The uniqueness of the representation ( ) of x by means of the family of scalars (a j ) j J allows us to define a family (x j ) j J of (continuous) functionals, associated to the basis (x j ) j J, by the formula x j (x) = a j, j J. Hence x j (x k) = δ jk, where δ jk is the Kronecker symbol, and x = j J x j (x)x j for every x X. The basis (x j ) j J is called symmetric if, for every sequence (j n ) in J the basic sequence (x jn ) is symmetric in the usual sense [15, p. 113]: for any sequence (r n ) in J the basic sequences (x jn ) and (x rn ) are equivalent. For details about uncountable unconditional and symmetric bases, we refer the reader to the monographs [11, 19]. Now let X denote a vector lattice, and let X + denote the cone of nonnegative elements of X. The lattice X is said to be [σ ] Dedekind complete if every nonempty [countable, resp.] subset A of X, with a b for all a A and some b X, has a supremum sup A in X. Two vector lattices X, Y are said to be order-isomorphic if there is a linear bijection R from X onto Y preserving finite suprema: R(sup{x 1, x 2 }) = sup{r(x 1 ), R(x 2 )} for all x 1, x 2 X; then R is referred to as an order isomorphism. If, additionally, X, Y are Banach lattices and R is a topological isomorphism, then we say that these spaces are order-topologically isomorphic. An element 0 < e X + is said to be discrete (or an atom) if, for every x X such that 0 < x e, there exists a real number λ > 0 such that x = λa. The symbol A X stands for the set of all discrete elements of X. A vector lattice X is said to be discrete (or atomic) whenever A X is a maximal disjoint system in X (i.e., the condition x e = 0 for all e A X implies x = 0). This is equivalent to the requirement that X is order-isomorphic to an order dense (i.e., cofinal) sublattice of the vector lattice of the form R Γ. Here, Γ can be chosen so that A X = {e γ : γ Γ} (see [3, Theorem 2.17, pp ]). An order ideal J of X is said to be [super-] order dense in X if, for every x X +, there is a [countable, resp.] subset A of J such that x = sup A; then x = sup([0, x] J). Without loss of generality we may assume that A is directed upward, and if A = {u n } n=1, then u n u n+1 x for all n. 4

5 An element x in a Banach lattice X is said to be order-continuous if for every net (x α ) in X with x x α 0, we have that x α 0. The order ideal X a, of all order-continuous elements of X, is referred to as the order-continuous part of X, and X a is norm-closed in X (see e.g. [22, p. 60]). The ideal X a is Dedekind complete (with the ordering inherited form X). If X = X a, we say that X has order continuous norm, or that X is order continuous (o.c.). It is easy to check that, for X a discrete Banach lattice, the order continuous part X a is an order dense ideal in X. In this case, we have X a = lin(a X ) (norm closure), and A X is an unconditional basis for X a. The classical Banach lattices X = l p (Γ), 1 p < are o.c., and for Y = l (Γ) we have Y a = c 0 (Γ). Here the set of the standard unit vectors e γ : γ Γ, forms a symmetric basis for the respective spaces X and Y a. For other examples see Section 5. The norm on a Banach lattice X is said to be Levi [resp., σ-levi] if every norm-bounded increasing net [resp., sequence] in X + has a supremum (cf., [3, Definition 9.3, p. 61]). In this case, we also say that X has the Levi [resp., σ-levi] property. As it is pointed out in [2] (see also [1]), this property appears in the literature under many different names. For example, the terms monotone complete norm and σ-levi norm are identical. It is known that, for every Banach lattice X, its dual X has the Levi property [17, Theorems and ], and it is obvious that every Banach lattice X with the Levi [resp., σ-levi] property is also [resp., σ-]dedekindcomplete. The following extension theorem will be applied in the proof of Theorem 4.2. Theorem 2.1. Let X, Y be two Banach lattices with the [σ-]levi property, and let J 1, J 2 be two [super-]order dense and norm-closed ideals of X and Y, respectively. Then every order isomorphism R from J 1 onto J 2 can be extended to an order isomorphism R from X onto Y. The extension is of the form R(x) = sup R([0, x] J 1 ), x X +. (1) Proof. By following the lines of the proof of Veksler s Theorem [4, Theorem 4.12], if we establish that, for every x X +, the supremum R(x) := sup R([0, x] J 1 ) exists in Y, then we can conclude R is an additive and positively homogeneous operator on X + which extends to a positive operator on X (denoted also by R) by the formula R(x) = R(x + ) R(x ). Let us note that then R(x + ) R(x ) = 0, whence R(x) = R( x ), thus R is an order homomorphism. The proof of (1) will be given only for the case when X and Y are σ-levi and the ideals J 1, J 2 are super-order dense in X and Y, respectively, as the proof of the other case is similar. Let us fix x X +. 5

6 Claim. If x = sup{u n : n N} for a nondecreasing sequence (u n ) in J + 1, then sup R([0, x] J 1 ) = sup n 1 R(u n ). The set A = {u n : n N} is topologically bounded in X (i.e. α n u n 0 for every sequence (α n ) c 0 ), thus from the continuity of R (see [4, Theorem 4.3]) it follows that the set {Ru n : n N} is topologically bounded and directed upward in Y, too. Since Y is σ-levi, the element b := sup{ru n : n N} exists in Y +. Moreover, for every u [0, x] J 1 we have u = u x = sup{u u n : n N}, thus from the order continuity of R we obtain Ru = sup{r(u u n ): n N} sup n 1 Ru n = b. It follows that b is the least upper bound of the set R([0, x] J 1 ), thus b = sup R([0, x] J 1 ). This proves our Claim, which shows that formula (1) is well defined. From the order density of J 1 in X we immediately obtain that the operator R is injective. Moreover, R is surjective. Indeed, let Y + y = sup{w n : n N} for a sequence (w n ) in J + 2, and set u n = R 1 (w n ), n = 1, 2,.... Then 0 u n u n+1 and u n R 1 w n R 1 y for all n 1. Since X is σ-levi, there exists x := sup n 1 u n in X. Now the Claim implies that R(x) = sup n 1 T u n = sup n 1 w n = y. Thus R maps X onto Y. 3 The l 1 -case The following property of l 1 (Γ)-spaces will be applied in the proof of our main result. It supplements Troyanski s result stated in Proposition 1.3 and seems to be new. Theorem 3.1. Let Γ and Θ be two uncountable sets, and let X be a nonseparable real Banach space with a symmetric basis {x t : t Θ}. If X can be injected continuously into l 1 (Γ), then the basis of X is equivalent to the standard basis of l 1 (Θ). The proof of the above theorem is based on the following combinatorial lemma [8, Lemma 2]: Lemma 3.2. Let {S t } t Θ be an uncountable family of (at most) countable subsets of a set S such that, for each s S, the set {t Θ: s S t } is (at most) countable. Then there exists a subset Θ of Θ such that card(θ ) = card(θ) and {S t } t Θ is a disjoint family. Proof of Theorem 3.1. Let (e γ ) γ Γ denote the standard symmetric basis of l 1 (Γ), and let (e γ) γ Γ be the dual family in l 1 (Γ), biorthogonal to (e γ ) γ Γ, and let R : X l 1 (Γ) be an injective and continuous mapping. Since the basis {x t : t Θ} is symmetric, we have that 0 < A := inf t Θ x t sup t Θ x t =: B < (see [12, Theorem 5], cf. [15, p. 114]). Set y t = Rx t, t Θ. Let us consider two cases: (a) there is γ 0 Γ such that e γ 0 (y t ) 0 for uncountably many t Θ, 6

7 (b) for every γ Γ, e γ(y t ) 0 only for countably many t Θ. In case (a), there is an uncountable subset Θ 0 of Θ and a positive number ε 0 such that e γ 0 (y t ) ε 0 for all t Θ 0. It follows that, for every family (a t ) t Θ0 of real numbers, the convergence of a series t Θ 0 a t x t implies the convergence of the series t Θ 0 a t. Since the elements of the symmetric basis {x t : t Θ} are norm-bounded (by B), the basic set {x t } t Θ0, and hence the basis {x t } t Θ, is an l 1 -basis. In case (b), we have the uncountable family F := {s(y t ) : t Θ} of (countable) supports of y t such that each subfamily F of F with F is countable. Now Lemma 3.2 implies that there is an uncountable subset Θ of Θ, with card(θ ) = card(θ), such that the elements of the subfamily {s(y t ) : t Θ } of F are pairwise disjoint. Since card(θ ) > ℵ 0, there is an uncountable subset Θ of Θ such that C := inf t Θ y t > 0. Hence the family {y t : t Θ } is norm-bounded from above (by B R ) and form below (by C) and consists of pairwise disjoint elements of l 1 (Γ), thus it is equivalent to the standard basis of l 1 (Θ ). It follows that the three series: t Θ a tx t, t Θ a ty t and t Θ a t converge simultaneously. But the basis {x t : t Θ} is symmetric, so the latter property implies the basis is equivalent to the standard basis of l 1 (Θ). The proof is complete. The next result shows that Proposition 1.4 cannot be extended to small l 1 -spaces, and that the restriction in our Theorem 4.2 to discrete Banach lattices X with card(a X ) 2 ℵ0 is, in a sense, natural. Theorem 3.3. Let D, G be two infinite sets of the cardinality 2 ℵ0. Then there is a continuous injection from (real or complex) l 1 (D) into l 1 (G). The result is independent on the Continuum Hypothesis. Proof. We assume without loss of generality that D G K, where K denotes the unit interval. Let (x n ) denote a fixed Schauder basis for C(K). Let us consider a linear continuous operator R from c 0 into C(K) of the form R((t n )) = (t n /n 2 )x n. n=1 The range of R is norm dense in C(K), thus its conjugate R is a continuous linear injection from C(K) into l 1 = c 0. Now let S denote the restriction of R to the atomic part of C(K), which is isometrically isomorphic to l 1 (K) (see e.g. [13, Theorem 7 on p. 50, and Theorem 9 on p. 52]). Hence S is a continuous injection S from l 1 (K) into l 1. Since the restriction S 1 of S to l 1 (G) is injective, and the natural embedding J of l 1 into l 1 (D) is injective too, the composition JS 1 is a continuous injection from l 1 (G) into l 1 (D), as required. 7

8 4 The main result Throughout this section, X denotes a discrete real Banach lattice with a fixed uncountable maximal set of discrete and pairwise disjoint elements A X = {x γ : γ Γ}. Recall that X a denotes the order continuous part of X. To shorten the text, we shall use the following notion. Definition 4.1. We say that X is a Drewnowski space, or a D-space, if it fulfils the following three conditions: (D1) either X is order continuous (i.e., X = X a ) or X has the σ-levi property, (D2) the ideal X a is super-order dense in X, (D3) the set A X is a symmetric basis for X a. By [16, p. 2], every nonseparable real Banach space X with an uncountable symmetric basis, endowed with the coordinatewise ordering, may be regarded as an o.c. Drewnowski space. The name Drewnowski space is justified by the Drewnowski s results cited in Section 1. Now we can present our main result in a concise form. Theorem 4.2. Let X and Y be two discrete Banach lattices such that both the sets, A X and A Y, are symmetric bases for X a and Y a, respectively, and that A X and A Y have the cardinality at least continuum. If there are continuous linear injections from X into Y and from Y into X (in particular, if X and Y are of the same linear dimension), then: (1.) the bases A X and A Y are permutatively equivalent: there is a bijection f : Γ Θ and an isomorphism R from X a onto Y a such that R(x γ ) = y f(γ), for every γ Γ; hence (2.) X a and Y a are order-topologically isomorphic. (3.) Moreover, if X and Y are non-order continuous D-spaces, then X and Y are order-topologically isomorphic: the above operator R extends to an order-topological isomorphism from X onto Y. It should be noted that the isomorphism of the spaces X and Y in Theorem 4.2 is obtained when X and Y are of the same order-continuity type: (a) From part (1.) (for the case X = X a and Y = Y a ) we immediately obtain a partial strengthening of Drewnowski s Proposition 1.4: if the symmetric bases of X and Y are of the cardinality at least continuum, then the conditions (i) (iv) are equivalent also in the case when one of the bases is of l 1 -type. (b) In the proof of part (3.) of the theorem we essentially apply the extension theorem 2.1 for X and Y non-order continuous. We do not know, however, whether the final conclusion of this part holds true also when X is o.c. and Y is not. 8

9 Before proceeding to the proof of the above theorem we shall present below two, somewhat surprising, properties of D-spaces. Notice first that the classical (real) space c 0 (Γ) is an o.c. D-space, and that, for every D-space X, with A X = {x γ : γ Γ}, the mapping X a γ Γ a γx γ (a γ ) γ Γ c 0 (Γ) is well defined (because the basis A X is norm bounded from below and from above) and it is a continuous injection. Moreover, the (real) space l c (Γ), of the elements of l (Γ) with a countable support, is a D-space, too, with (l c (Γ)) a = c 0 (Γ). Corollary 4.3. Let X be a D-space with card(a X ) = card(γ). Then: (i) If the spaces X a and c 0 (Γ) are not isomorphic (as Banach spaces), then there is no continuous injection from c 0 (Γ) into X a ; (ii) If X is not o.c. and the spaces X a and c 0 (Γ) are isomorphic, then the Banach lattices X and l c (Γ) are order-topologically isomorphic. Parts (i) and (ii) of the corollary are direct consequences of Theorem 4.2 and the extension theorem 2.1, respectively. Proof of Theorem 4.2. Part (1.). Let T : X Y and S : Y X be continuous linear injections. Step 1. Set f γ := T x γ, γ Γ. Since T is injective, f γ 0 for all γ. Let y t denote the functional associated to y t, t Θ. We consider two cases: (i) there is t 0 Θ such that y t 0 (f γ ) 0 for uncountably many γ; (ii) for every t Θ, y t (f γ ) 0 only for countably many γ. In case (i), repeating the proof of case (a) in the proof of Theorem 3.1 we obtain that the basis A X of X a is equivalent to the standard basis of l 1 (Γ), i.e., X a is order-topologically isomorphic to l 1 (Γ). In particular, X a is a KB-space (i.e., every increasing norm bounded sequence in X + is norm convergent to an element of X). Since X a is super-order dense in X, the latter implies that X a is topologically dense in X, whence X = X a, because X a is norm-closed in X. Thus, in case (i), X is order-topologically isomorphic to l 1 (Γ), and so, by the hypothesis of our theorem, we may assume without loss of generality that S maps Y, whence Y a too, into l 1 (Γ) injectively. By Theorem 3.1, Y a is order isomorphic to l 1 (Θ), and by the arguments as above, Y = Y a ; thus we may also assume that Y = l 1 (Θ). Now the injectivity of T and S along with the hypothesis that card(a X ), card(a Y ) 2 ℵ0 imply that card(γ) = card(θ). Hence the Banach lattices X = X a and Y = Y a are order-topologically isomorphic (to a big l 1 -space). Similar arguments employed to the case (i ) there is γ 0 Γ such that x γ 0 (g t ) 0 for uncountably many t Θ, where g t = Sy t, t Θ, also give us that X = X a and Y = Y a are order-topologically isomorphic to an l 1 -space. 9

10 It thus remains to consider case (ii) along with its counterpart: (ii ) for every γ Γ, x γ(g t 0 only for countably many t Θ, where g t = S(y t ). Step 2. Assume (ii) and (ii ). By Lemma 3.2, there are uncountable subsets Γ 0 and Θ 0 of Γ and Θ, respectively, such that the elements of either of the sets {T (x γ ): γ Γ 0 } and {S(y t ): t Θ 0 } are pairwise disjoint. Now we mimic a part of Drewnowski s proof of Case 2 of [8, Theorem]. Let a series γ Γ 0 a γ x γ be convergent in X. Then the series γ Γ 0 a γ T (x γ ) is convergent in Y ; equivalently (as the elements of {T (x γ ) : γ Γ 0 } are pairwise disjoint), the series γ Γ 0 a γ T (x γ ) is convergent in Y. (2) We have that for every u γ := T (x γ ) Y + \ {0}, where γ Γ 0, there is yt(γ) such that yt(γ) (u γ) > 0 (i.e., t(γ) s(u γ )) and u γ yt(γ) (y γ) y t(γ). Now from (2) it follows that the series a γ yt(γ) (u γ) y t(γ) (3) γ Γ 0 converges in Y, too. But the set Γ 0 is uncountable, so we may assume that the number inf γ Γ0 y t(γ) (u γ) is positive, thus the convergence of the series (3) implies the convergence of the series γ Γ 0 a γ y t(γ). Since the elements y t(γ), γ Γ 0, are pairwise disjoint we have that the basic set {x γ : γ Γ 0 } dominates the basic set {y t(γ) : γ Γ 0 } {y t : t Θ} = A Y, with card(γ) = card(γ 0 ) card(θ). But the bases are symmetric, so the latter implies that for every two sequences {γ n } Γ and {t n } Θ the basic sequence {x γn } dominates the basic sequence {y tn }. Similarly, there is an uncountable basic subset of A Y = {y t : t Θ} of the same cardinality as Θ, dominating an uncountable basic subset of A Y = {x γ : γ Γ}, and card(θ) card(γ). Hence (similarly as above), the basic sequence {y tn } dominates the basic sequence {x γn }, so that the sequences are equivalent. We thus have obtained that the sets Γ and Θ are of the same cardinality and that the bases A X and A Y of X a and Y a, respectively, are equivalent. This means that there is a bijection f from Γ onto Θ such that, for every family (a γ ) γ Γ of real numbers, both the series, γ Γ a γx γ and γ Γ a γy f (γ), converge (topologically) simultaneously. It follows that the linear operator R from X a onto Y a such that R(x γ ) = y f (γ), for all γ Γ, is well defined and that R is a bijection. Because the orderings of X a and Y a are coordinatewise, the operators R and R 1 are positive, whence continuous [4, Theorem 12.3]. The proof of part (1.) of our theorem is complete. Part (2.). Both the operators R and R 1, defined above, are positive, hence order isomorphisms [4, Theorem 7.3]. Thus R is an order-topological isomorphism from X a onto Y a. 10

11 Part (3.). This part follows from Theorem 2.1 applied to part (2.). Remark 4.4. An inspection of Step 1 of the proof of Theorem 4.2 shows that if X a (or Y a ) is not an l 1 -space, then we may replace the request for A X and A Y to be of the cardinality at least continuum by the weaker hypothesis: card(a X ), card(a X ) > ℵ 0. 5 Applications to Orlicz spaces lϕ(γ) In this section, we present a method of constructing D-spaces from small Banach sequence lattices with the σ-levi property. The construction is similar to Drewnowski s construction of order continuous D-spaces [8, p. 158]. Let W = (W, W ) be a σ-dedekind complete Banach sequence lattice, i.e., W is an order ideal of the lattice R N such that, if x = (a n ) W and b n a n for all n s, then y = (b n ) W and y W x W. Let us assume that W is a symmetric sequence space: for every x = (a n ) W and every permutation π of the integers, we have (s 1 ) the element π(x) := (a π(n) ) lies in W, and (s 2 ) x = π(x). These two conditions imply that the sequence (e n ) of the standard unit vectors is a symmetric basis of the order continuous part W a of W, and that, if x = (a n ) W, then a n e n = a n e π(n) = a π (n) e kn (4) W W W n=1 n=1 for every permutation π and every sequence (k n ) of pairwise distinct positive integers [15, pp ]. Now we set W (Γ) := {x R Γ : s(x) is either finite, or countable infinite and x s(x) W } (recall that s(x) denotes the support of x). Then W (Γ), endowed with the coordinatewise ordering, is a σ-dedekind complete linear lattice. If x = (a γ ) γ Γ W (Γ) has an infinite support, we set x := x s(x) W. If s(x) =, we set x = 0, and for s(x) finite (nonempty): s(x) = {γ 1,..., γ k }, we complete s(x) with any infinite sequence (of pairwise distinct elements) {γ k+1,...} Γ and we set x := (a γn ) W, where a γn = 0 for n k + 1. By property 4, (W (Γ), ) is a Banach lattice, and it is easy to check that its order continuous part (W (Γ)) a equals W a (Γ). Moreover, (p 1 ) if W is not o.c. and has the σ-levi property, then W (Γ) is also of this type (obvious); n=1 11

12 (p 2 ) if U is another symmetric sequence space and the spaces W a (Γ) and U a (Γ) are isomorphic, then the (symmetric) bases of W a and U a are equivalent (by Proposition 1.2). In particular, for W = l we obtain the D-space l c (Γ) with (W (Γ)) a = W a (Γ) = c 0 (Γ). Now we come to a larger class of non-o.c. symmetric sequence spaces with the Levi property. An Orlicz function ϕ is a non-negative, non-decreasing, convex function defined on [0, ) and satisfying ϕ(0) = 0. We define the convex modular functional ϱ ϕ : R N [0, ] by the formula ϱ ϕ (x) = n=1 ϕ( a n ), x = (a n ). The Orlicz space l ϕ is a Banach sequence lattice of the form endowed with the norm l ϕ = {x R N : ϱ ϕ (x/λ) <, for some λ > 0} x ϕ = inf{λ > 0: ϱ ϕ (x/λ) 1}. It is obvious that l ϕ is a symmetric sequence space, and it is known [21] that the space h ϕ := {x R N : ϱ ϕ (x/λ) <, for all λ > 0} is the order continuous part of l ϕ. We have that l ϕ = h ϕ if and only if ϕ fulfils the so-called 2 -condition at 0: there are positive numbers t 0 and K such that ϕ(2t) Kϕ(t) for all t [0, t 0 ] (see e.g., [15, Proposition 4.a.4]); then we write ϕ 2. It is easy to verify that the Orlicz function ϕ of the form ϕ(x) = x 2 e 1/x, for x > 0, and ϕ(0) = 0, does not satisfy the 2 -condition at 0. For other examples of Orlicz functions ϕ such that ϕ 2 see e.g. [7]. In the latter case, l ϕ is a dual Banach lattice [15, Proposition 4.b.1], thus it has the Levi property [17, Theorems and ]. Summing up, if ϕ 2, then W := l ϕ is a symmetric non-o.c. sequence space with the Levi property. By property (p 1 ), if ϕ 2 then l ϕ (Γ) is a non-o.c. D-space and its order continuous part (l ϕ (Γ)) a equals h ϕ (Γ). Hence, from Theorem 4.2, property (p 2 ), and [15, Proposition 4.a.5] we obtain Corollary 5.1. Let Γ, Θ be two sets of the cardinality at least continuum, and let ϕ, ψ be two Orlicz functions not fulfilling the 2 -condition at 0. Then the following four conditions are equivalent: (i) There are continuous linear injections from l ϕ (Γ) into l ψ (Θ) and vice versa. (ii) The Banach lattices l ϕ (Γ) and l ψ (Θ) (as well as h ϕ (Γ) and h ψ (Θ)) are order-topologically isomorphic. 12

13 (iii) The natural unit vector bases of h ϕ and h ψ are equivalent. (iv) The functions ϕ and ψ are equivalent at 0: there exist constants k, K > 0 and t 0 > 0 such that for all t [0, t 0 ] K 1 ϕ(k 1 t) ψ(t) Kϕ(kt). We end this section with the following observation. If ϕ is an Orlicz function with ϕ 2, then l ϕ contains an isomorphic copy of l (see [15, Proposition 4.a.4]), so l ϕ (R) does. Moreover, the identity embedding of l ϕ (R) into the D-space l c (R) is continuous (e.g., because positive). We claim that If ϕ is nondegenerate (i.e., ϕ does not vanish on an interval [0, s] for some s > 0), then l ϕ (R) contains only isomorphic copies of l and not l c (R). In particular, there is no extension of any isomorphic embedding l l ϕ (R) to an isomorphism (and even to a continuous injection) l (R) l ϕ (R). Indeed, if there was a continuous injection from l c (R) into l ϕ (R) then, by Theorem 4.2, the bases of c 0 (R) = (l c (R)) a and h ϕ (R) = (l ϕ (R)) a would be equivalent. By property (p 2 ), we would have h ϕ = c 0, thus ϕ would be degenerate; a contradiction. References [1] Y. A. Abramovich and C. D. Aliprantis, Positive operators, Handbook of the geometry of Banach spaces, Vol. I, pp , North-Holland, Amsterdam, (2001). [2] Y. A. Abramovich and A. W. Wickstead, When each continuous operator is regular. II, Indag. Math. 8 (1997) [3] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces, Pure and Applied Mathematics Series, vol. 76, Academic Press Inc., New York, [4] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Pure and Applied Mathematics Series, vol. 119, Academic Press Inc., Orlando, London, [5] S. Banach, Théorie des opérations linéaires, Monografje Matematyczne, Warszawa, [6] S. Banach and S. Mazur, Zur Theorie der linearen Dimensionen, Studia Math., 4 (1933), [7] S. Chen, Geometry of Orlicz spaces, Dissertationes Math., 356 (1995). [8] L. Drewnowski, On symmetric bases in non-separable Banach spaces, Studia Math., 85 (1987), [9] L. Drewnowski, On uncountable unconditional bases in Banach spaces, Studia Math., 90 (1988),

14 [10] C. Finol and M. Wójtowicz, The structure of nonseparable Banach spaces with uncountable unconditional bases, Rev. R. Acad. Cien. Serie A. Mat., Vol. 99 no. 1 (2005), [11] P. Hájek, V. Montesinos, Santalucía, J. Vanderwerff and V. Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 26, Springer-Verlag, New York, [12] M. I. Kadec and A. Pe lczyński, Bases, lacunary sequences and complemented subspaces in the spaces L p, Studia Math. 21 (1962), [13] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer Verlag, Berlin, [14] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics 338, Springer-Verlag, Berlin, [15] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer- Verlag, Berlin, [16] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer- Verlag, Berlin, [17] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin, [18] R. E. A. C. Paley, Some theorems on abstract spaces, Bull. Amer. Math. Soc. 42 (1936), [19] I. Singer, Bases in Banach spaces II, Springer-Verlag, Berlin, [20] S. Troyanski, On nonseparable Banach spaces with a symmetric basis, Studia Math., 53 (1975), [21] W. Wnuk, On the order-topological properties of the quotient space L/L A, Studia Math., 79 (1984), [22] W. Wnuk, Banach Lattices with Order Continuous Norms, Polish Scientific Publishers PWN, Warszawa, [23] A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, Address: Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000, Caracas, 1080-A, Venezuela mago@usb.ve Address: Instytut Matematyki Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, Bydgoszcz, Poland. mwojt@ukw.edu.pl 14